Output from radio feeding both a speaker and microphone input

The TRRS connectors are wired like this, in order:

• Tip: Left output
• Ring: Right output
• Ring: Ground/return
• Sleeve: Microphone input

That way the headphones will work plugged into a normal TRS jack, too.

The microphone input is meant to connect to an electret mic with a circuit like this:

The bias voltage, resistor, and DC-blocking capacitor are all inside the phone. So just connecting a signal source like a computer's sound card to the microphone input (with ground to the ground ring) should work.

If your sound source's DC output resistance is too low (normally a good thing), the phone could be interpreting it as a continuous button press (Answer or Play/Pause), because the headset button shorts the mic line's bias resistor to ground. Does it do anything to indicate that it thinks a button is being pressed? If this is the case, just put an electrolytic capacitor in series with your source, like 10 µF to 100 µF between your source and the microphone input. This will block the DC current from the bias resistor while allowing the AC signal to pass. Larger capacitance will have better low-frequency response.

For instance, if the input impedance of the phone is 2.2 kΩ, 10 µF will have a -3 dB cutoff frequency of 7 Hz, which is plenty.

What you want to do should work, as long as your sound card really has stereo microphone inputs. Most will have stereo "line" inputs and at least some kind of microcphone input, but you need to check whether that mic input is stereo or not. If it is, just hook up the two mics to the common ground and the other side of each mic to one of the two input channels. Make sure to use shielded audio cable.

If you don't have a stereo mic input, then it would be better to use a separate preamp so that each signal is amplified to "line" level, then put the two line level signals into the line level input of the sound card. In that case you have to be very careful about ground loops since the preamp will be separately powered. Plug it in as close as possible to the same place the PC is plugged in to, and don't hook up a separate deliberate ground to it other than thru the line in connection of the sound card.